Upper bounds on Fourier entropy

Chakraborty, Sourav; Kulkarni, Raghav; Lokam, Satyanarayana V.; Saurabh, Nitin

doi:10.1016/j.tcs.2016.05.006

Cited by 15 publications

(26 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To prove the FEI conjecture for a function f (or, any class of Boolean functions) one needs to provide an upper bound on the Fourier entropy of f and a matching (up to a constant factor) lower bound on the influence of f . For upper bounding the Fourier entropy of a LTF we crucially use the following theorem proved in [18].…”

Section: Our Proof Techniquementioning

confidence: 99%

Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

Chakraborty

Karmalkar

Kundu

et al. 2018

LATIN 2018: Theoretical Informatics

Self Cite

View full text Add to dashboard Cite

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function f : {+1, −1} n → {+1, −1}, the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a "random" linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on [−1, 1] and Normal distribution.

show abstract

Section: Our Proof Techniquementioning

confidence: 99%

Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

Chakraborty

Karmalkar

Kundu

et al. 2018

LATIN 2018: Theoretical Informatics

Self Cite

View full text Add to dashboard Cite

show abstract

“…As mentioned in the introduction, it was shown by [1,9] that read-once formulas satisfy Conjecture 1.1 with the constant C ≤ 10.…”

Section: Disjoint Compositionmentioning

confidence: 99%

“…Conjecture 1.1 was verified for various families of Boolean functions (e.g., symmetric functions [10], random functions [3], read-once formulas [1,9], decision trees of constant average depth [11], read-k decision trees for constant k [11]) but is still open for the class of general Boolean functions.…”

Section: Conjecture 11 ([4]mentioning

confidence: 99%

“…to prove Pr [g m ] = 1 2 , it suffices to verify the following (see Appendix A for the calculation):…”

Section: Sequences Of Balanced Monotone Functionsmentioning

confidence: 99%

“…Remark. The formula (1) shows that every monotone decision list, i.e., a monotone decision tree consisting of a single path, is isomorphic to a lexicographic function.…”

Section: Lexicographic Functionsmentioning

confidence: 99%

See 2 more Smart Citations

Improved Lower Bounds for the Fourier Entropy/Influence Conjecture via Lexicographic Functions

Hod

2017

Preprint

View full text Add to dashboard Cite

Every Boolean function can be uniquely represented as a multilinear polynomial. The entropy and the total influence are two ways to measure the concentration of its Fourier coefficients, namely the monomial coefficients in this representation: the entropy roughly measures their spread, while the total influence measures their average level. The Fourier Entropy/Influence conjecture of Friedgut and Kalai from 1996 states that the entropy to influence ratio is bounded by a universal constant C.Using lexicographic Boolean functions, we present three explicit asymptotic constructions that improve upon the previously best known lower bound C > 6.278944 by O'Donnell and Tan, obtained via recursive composition. The first uses their construction with the lexicographic function ℓ 2/3 of measure 2/3 to demonstrate that C ≥ 4 + 3 log 4 3 > 6.377444. The second generalizes their construction to biased functions and obtains C > 6.413846 using ℓ Φ , where Φ is the inverse golden ratio. The third, independent, construction gives C > 6.454784, even for monotone functions.Beyond modest improvements to the value of C, our constructions shed some new light on the properties sought in potential counterexamples to the conjecture.Additionally, we prove a Lipschitz-type condition on the total influence and spectral entropy, which may be of independent interest.

show abstract